Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:0809.1730 · Physical Review B

Dynamical correlations in the Sherrington-Kirkpatrick model in a transverse field

Kazutaka Takahashi, Koujin Takeda

Published Nov 17, 2008• 7 citations
arXivHTMLPDF

Abstract

We calculate the real-time-correlation function of the Sherrington-Kirkpatrick spin-glass model in a transverse field. Using a careful analysis of the perturbative expansion of the functional-integral representation, we derive the asymptotic form of the correlation function. In contrast to the previous works, we find for large transverse field a power-law decay of the correlation with time $t$ as ${t}^{\ensuremath{-}3/2}$ at zero temperature and ${t}^{\ensuremath{-}2}$ at infinite temperature. At the small field region, we also find a significant change in the correlation which comes from the structural difference between two paramagnetic phases.

DOI
10.1103/physrevb.78.174415

Related papers

A dynamic theory of transverse freezing in the Sherrington-Kirkpatrick Ising model

T. K. Kopeć

The effect of a transverse field Gamma on spin-glass freezing in the Sherrington-Kirkpatrick version of the quantum Ising model is studied by using the thermo-field-dynamic (TFD) method (the real-time finite-temperature quantum-field theory) due to Takahashi and Umezawa as a substitute for the 'n-replica trick'. An exact effective Lagrangian for the average dynamic causal and response functions is derived by using the functional-integral approach. Furthermore, the properties of the Lagrangian are studied in the mean-field limit by considering the infinite-range random exchange. In this limit the self-consistent equations for the matrix components of the TFD causal Green function are derived with the effective thermal Hamiltonian containing the dynamic spin self-interaction. The static properties of this equation are studied above and below the freezing temperature T c by extracting the time-delocalised parts of dynamic autocorrelations. The self-consistent equations for the spin-glass order parameter and local susceptibility are obtained. The classical ( Gamma to 0) and quantum (T to 0) limits are discussed. The crucial role of the choice of the thermal-state condition for the spin-glass problem and its striking similarity to the non-equilibrium formulation of the TFD are pointed out.

Journal of Physics C Solid State Physics•Dec 30, 1988•31 citations

Complete replica solution for the transverse field Sherrington-Kirkpatrick spin glass model with continuous-time quantum Monte Carlo method

Annamária Kiss, Gergely Zaránd, Izabella Lovas

We construct a complete numerically exact solution of a mean-field quantum spin glass model---the transverse field Sherrington-Kirkpatrick model---by implementing a continuous-time quantum Monte Carlo method in the presence of full replica symmetry breaking. We extract the full numerically exact phase diagram, displaying a glassy phase with continuous replica symmetry breaking at small transverse fields and low temperatures. A paramagnetic phase emerges once thermal and quantum fluctuations melt the spin glass. We characterize both phases by extracting the order parameter as well as the static and dynamical local spin susceptibilities. The static susceptibility shows a plateau in the glassy phase, but it remains smooth across the phase boundary. For the imaginary part of the dynamical susceptibility, we find an Ohmic, i.e., linear in $\ensuremath{\omega}$, scaling for small frequencies $\ensuremath{\omega}$, with a slope independent of the transverse field. These results compare qualitatively well with ac susceptibility measurements on a dipole-coupled three-dimensional Ising magnet---the ${\mathrm{LiHo}}_{x}{\mathrm{Y}}_{1\ensuremath{-}x}{\mathrm{F}}_{4}$ compound---in a transverse magnetic field. Our work provides a general framework for the exact numerical solution of mean-field quantum glass models, and it constitutes an important step towards understanding glassiness in realistic systems.

arXiv:2306.07337•Physical review. B./Physical review. B•Jan 25, 2024•9 citations

Effect of a transverse field on the spin glass freezing in the Sherrington-Kirkpatrick model

H Ishii, Tetsuya Yamamoto

In the Sherrington-Kirkpatrick (SK) model the effect of a transverse field Delta on the spin glass freezing is studied on the basis of the theory of Thouless, Anderson and Palmer (1977). It is shown from the logarithmic singular point of the free energy in the high-temperature region that T c decreases with increasing Delta . To be consistent with this result the mean-field equation is set up for T<T c . Due to the correlation effect spin flips associated with the transverse field are unlikely to occur at low temperatures and the system eventually undergoes the spin glass transition irrespective of the magnitude of Delta , which is characteristic of the SK model.

Journal of Physics C Solid State Physics•Nov 30, 1985•88 citations

Quantump-state Potts spin glass in a transverse field: Dynamical correlations and first-order phase transition

T. K. Kopeć, G. Büttner

The p-component quantum Potts spin-glass model is considered in the mean-field approximation via the thermofield-dynamics (TFD) method. Quantum effects appear as the action of a transverse field \ensuremath{\Gamma} which represents the tunneling frequency between different discrete Potts states. Applying the Sherrington-Kirkpatrick infinite-interaction-range scheme in the TFD formalism, we find a continuous spin-glass transition for p\ensuremath{\le}6 and discontinuous for p>6 at the field-dependent critical line ${\mathit{T}}_{\mathit{c}}$(\ensuremath{\Gamma}), as demonstrated in the jump of the temperature dependence of the spin-glass order parameter. Interestingly, in contrast to the statics the dynamical theory leads to a continuous slowing down as ${\mathit{T}}_{\mathit{c}}$(\ensuremath{\Gamma}) is approached from the ergodic side, indicating the existence of the infinitely-long-lived metastable states.

Physical review. B, Condensed matter•Jul 1, 1992•0 citations

A perturbation expansion for the Sherrington-Kirkpatrick model with a transverse field

Tetsuya Yamamoto, H Ishii

In the Sherrington-Kirkpatrick model the effect of a transverse field Delta on spin glass freezing is studied. After the static approximation of the replica theory has been analysed, the nonlinear susceptibility is calculated by the perturbation method with respect to exchange coupling J ij up to the tenth order. The freezing temperature and the critical index of the nonlinear susceptibility are evaluated. The result shows that the system remains paramagnetic for Delta >1.506 J (the variance of (J ij ) is J 2 /N), which corrects the conjecture made before by the present authors. It is also shown that the static approximation in the replica theory overestimates T c for a large value of Delta .

Journal of Physics C Solid State Physics•Dec 20, 1987•86 citations

Free energy of the quantum Sherrington–Kirkpatrick spin-glass model with transverse field

Arka Adhikari, Christian Brennecke

We consider the quantum Sherrington–Kirkpatrick (SK) spin-glass model with a transverse field and provide a formula for its free energy in the thermodynamic limit, valid for all inverse temperatures β > 0. To characterize the free energy, we use the path integral representation of the partition function and approximate the model by a sequence of finite-dimensional vector-spin glasses with Rd-valued spins. This enables us to use the results of Panchenko who generalized [Ann. Probab. 46(2), 829–864 (2018); ibid, 46(2), 865–896 (2018)] the Parisi formula to classical vector-spin glasses. As a consequence, we can express the thermodynamic limit of the free energy of the quantum SK model as the d → ∞ limit of the free energies of the d-dimensional approximations of the model.

arXiv:1912.13041•Journal of Mathematical Physics•Aug 1, 2020•11 citations
We calculate the real-time-correlation function of the Sherrington-Kirkpatrick spin-glass model in a transverse field. Using a careful analysis of the perturbative expansion of the functional-integral representation, we derive the asymptotic form of the correlation function. In contrast to the previous works, we find for large transverse field a power-law decay of the correlation with time ttt as t\ensuremath−3/2{t}^{\ensuremath{-}3/2}t\ensuremath−3/2 at zero temperature and t\ensuremath−2{t}^{\ensuremath{-}2}t\ensuremath−2 at infinite temperature. At the small field region, we also find a significant change in the correlation which comes from the structural difference between two paramagnetic phases.
The effect of a transverse field Gamma on spin-glass freezing in the Sherrington-Kirkpatrick version of the quantum Ising model is studied by using the thermo-field-dynamic (TFD) method (the real-time finite-temperature quantum-field theory) due to Takahashi and Umezawa as a substitute for the 'n-replica trick'. An exact effective Lagrangian for the average dynamic causal and response functions is derived by using the functional-integral approach. Furthermore, the properties of the Lagrangian are studied in the mean-field limit by considering the infinite-range random exchange. In this limit the self-consistent equations for the matrix components of the TFD causal Green function are derived with the effective thermal Hamiltonian containing the dynamic spin self-interaction. The static properties of this equation are studied above and below the freezing temperature T c by extracting the time-delocalised parts of dynamic autocorrelations. The self-consistent equations for the spin-glass order parameter and local susceptibility are obtained. The classical ( Gamma to 0) and quantum (T to 0) limits are discussed. The crucial role of the choice of the thermal-state condition for the spin-glass problem and its striking similarity to the non-equilibrium formulation of the TFD are pointed out.
We construct a complete numerically exact solution of a mean-field quantum spin glass model---the transverse field Sherrington-Kirkpatrick model---by implementing a continuous-time quantum Monte Carlo method in the presence of full replica symmetry breaking. We extract the full numerically exact phase diagram, displaying a glassy phase with continuous replica symmetry breaking at small transverse fields and low temperatures. A paramagnetic phase emerges once thermal and quantum fluctuations melt the spin glass. We characterize both phases by extracting the order parameter as well as the static and dynamical local spin susceptibilities. The static susceptibility shows a plateau in the glassy phase, but it remains smooth across the phase boundary. For the imaginary part of the dynamical susceptibility, we find an Ohmic, i.e., linear in $\ensuremath{\omega}$, scaling for small frequencies $\ensuremath{\omega}$, with a slope independent of the transverse field. These results compare qualitatively well with ac susceptibility measurements on a dipole-coupled three-dimensional Ising magnet---the ${\mathrm{LiHo}}_{x}{\mathrm{Y}}_{1\ensuremath{-}x}{\mathrm{F}}_{4}$ compound---in a transverse magnetic field. Our work provides a general framework for the exact numerical solution of mean-field quantum glass models, and it constitutes an important step towards understanding glassiness in realistic systems.
In the Sherrington-Kirkpatrick (SK) model the effect of a transverse field Delta on the spin glass freezing is studied on the basis of the theory of Thouless, Anderson and Palmer (1977). It is shown from the logarithmic singular point of the free energy in the high-temperature region that T c decreases with increasing Delta . To be consistent with this result the mean-field equation is set up for T<T c . Due to the correlation effect spin flips associated with the transverse field are unlikely to occur at low temperatures and the system eventually undergoes the spin glass transition irrespective of the magnitude of Delta , which is characteristic of the SK model.
The p-component quantum Potts spin-glass model is considered in the mean-field approximation via the thermofield-dynamics (TFD) method. Quantum effects appear as the action of a transverse field \ensuremath{\Gamma} which represents the tunneling frequency between different discrete Potts states. Applying the Sherrington-Kirkpatrick infinite-interaction-range scheme in the TFD formalism, we find a continuous spin-glass transition for p\ensuremath{\le}6 and discontinuous for p>6 at the field-dependent critical line Tc{\mathit{T}}_{\mathit{c}}Tc​(\ensuremath{\Gamma}), as demonstrated in the jump of the temperature dependence of the spin-glass order parameter. Interestingly, in contrast to the statics the dynamical theory leads to a continuous slowing down as Tc{\mathit{T}}_{\mathit{c}}Tc​(\ensuremath{\Gamma}) is approached from the ergodic side, indicating the existence of the infinitely-long-lived metastable states.
In the Sherrington-Kirkpatrick model the effect of a transverse field Delta on spin glass freezing is studied. After the static approximation of the replica theory has been analysed, the nonlinear susceptibility is calculated by the perturbation method with respect to exchange coupling J ij up to the tenth order. The freezing temperature and the critical index of the nonlinear susceptibility are evaluated. The result shows that the system remains paramagnetic for Delta >1.506 J (the variance of (J ij ) is J 2 /N), which corrects the conjecture made before by the present authors. It is also shown that the static approximation in the replica theory overestimates T c for a large value of Delta .
We consider the quantum Sherrington–Kirkpatrick (SK) spin-glass model with a transverse field and provide a formula for its free energy in the thermodynamic limit, valid for all inverse temperatures β > 0. To characterize the free energy, we use the path integral representation of the partition function and approximate the model by a sequence of finite-dimensional vector-spin glasses with Rd-valued spins. This enables us to use the results of Panchenko who generalized [Ann. Probab. 46(2), 829–864 (2018); ibid, 46(2), 865–896 (2018)] the Parisi formula to classical vector-spin glasses. As a consequence, we can express the thermodynamic limit of the free energy of the quantum SK model as the d → ∞ limit of the free energies of the d-dimensional approximations of the model.